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Mirrors > Home > MPE Home > Th. List > abeq1 | Structured version Visualization version GIF version |
Description: Equality of a class variable and a class abstraction. Commuted form of abeq2 2922. (Contributed by NM, 20-Aug-1993.) |
Ref | Expression |
---|---|
abeq1 | ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq2 2922 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
2 | eqcom 2805 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ 𝐴 = {𝑥 ∣ 𝜑}) | |
3 | bicom 225 | . . 3 ⊢ ((𝜑 ↔ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) | |
4 | 3 | albii 1821 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
5 | 1, 2, 4 | 3bitr4i 306 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: rabeqc 3626 disjOLD 4356 dm0rn0 5759 dffo3 6845 dfsup2 8892 rankf 9207 fmla0xp 32743 dfon3 33466 dfiota3 33497 scottabf 40948 dffo3f 41806 |
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